Green's Theorem
Green's Theorem is a result in real analysis. It is a special case of Stokes' Theorem.
Statement
Let be a bounded subset of
with positively
oriented boundary
, and let
and
be functions with
continuous partial derivatives mapping an open set containing
into
. Then
Proof
It suffices to show that the theorem holds when is a square,
since
can always be approximated arbitrarily well with
a finite collection of squares.
Then let be a square with vertices
,
,
,
, with
and
. Then
Now, by the Fundamental Theorem of Calculus,
and
Hence
as desired.